Tuesday, May 11, 2004

Last week, I finally got around to joining Friendster, after several friends told me it was worth checking out. Can’t say I’m blown away, but it is moderately interesting. It was probably more interesting back in the early days, when lots of people were trying it out for the first time and trying to expand their “friend networks” as much as possible. I sense that there is now a lull, except perhaps for whole groups of friends who are all just trying out Friendster for the first time.

Anyway, I was thinking about the mathematical properties and assumptions underlying the Friendster system. The implicit assumption is that friendship is a quasi-transitive operator: if A is friends with B and B is friends with C, then A is (sort of) friends with C. But the extent of friendship attenuates with the degrees of separation. We could imagine a friendship function F(n), where n is the degrees of separation. F(n) might have a form like so:

F(n) = k^n

where k is some real number in the interval (0, 1]. If k = 1, then friendship is perfectly transitive: a friend-of-a-friend-of-a-friend is just like a friend. On the other hand, if k < 1, then friendship attenuates with distance.

And then I thought, what if someone started a new website called “Fiendster”? On this website, you would create a network of your enemies. You’d send your enemies Fiendster invitations: “Glen has invited you to be his enemy. Do you accept?” And then you could check out your enemies’ lists of enemies to find potential allies, working on the theory that “the enemy of my enemy is my friend.” Mathematically, in this case k lies in the interval [-1, 0). For instance, we might have F(n) = (-.9)^n. This would mean that my enemy is my enemy to an extent of -.9, my enemy’s enemy is my ally to an extent of .81 (notice the sign change), my enemy’s enemy’s enemy is my enemy to an extent of -.729, and so on. Mathematical difficulties might arise if someone were both my enemy and the enemy of my enemy, because that person would be assigned both a positive and a negative number. In such cases, perhaps you would have to choose which of your enemies is the greater enemy. Or not – maybe the mathematical contradiction maps a real contradiction in human relationships.